maths primes

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Threshold Mathematics V4.01

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What's missing that makes maths such a turn-off for so manyand fascinating for a few?

And whats the deeper significance ofZero, One, Pi,the Golden Ratio, the Fibonacci series, the Primes,

Negative and Imaginary numbers... for a start?

CONTENTS

Prologue

1. Threshold mathematics is...

2. EMU numbers / Threshold numbers

3. The Primes and Riemann

4. Beyond Mathematics: Riemann's Significance

5. The Aether and Mathematics

Postscript

Pathway (Initiatives) Ltd 2006-2007PO Box 1181 Tring HP23 5WJ England

www.aetheraware.org

Re-publishing this paper in full and unedited will probably be permittedsubject to applying for and receiving permission in writing

from the original publisher (above).

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Prologue

Professor Marcus du Sautoy, in his book The Music of the Primes, laid it all out withmasterful clarity and flair - the big tease about the Primes, that is. All about to be

revealed... but then... hold on. Not quite yet. Ever closer but never quite getting there.What in maths is called asymptotic: the line and the curve that are converging ever closer,yet never quite make contact. And that seems to symbolize the fascinating story of theprimes, 'the most tantalising enigma', the biggest unresolved mystery in mathematics.

The essence of it has been the lack so far of (a) any meaningful explanation of thedistribution of the prime numbers and (b) as an extension of this, any kind of proof ordisproof of the now legendary Riemann Hypothesis.

Threshold Mathematics has looked at both in a radically different way and asked: Couldthese long running failures all along have been the inevitable consequence of asking the

wrong questions, based on false assumptions, themselves passed on unquestioninglythrough generations? And since the desired solutions have not been found after extensiveexpert analysis of the numbers, would it be more fruitful to look at the bigger context?

This work is, therefore, as much about how we have come to think about quantifying andnumbers as it is about mathematics itself along with its various conundrums. It wasmathematician, Kurt Godel, in the 1930s who showed that there are some statementswhich can neither be shown to be true nor untrue within mathematics, news which wasapparently not at all well received at the time by many leading mathematicians.Other broader questions arise. Could there be a vital factor missing that makes

mathematics such a turn-off for so many, yet deeply fascinating in its rationality,complexity and beauty for a small minority? How significant is its exclusion of thoseessential livingqualities we experience as willingand feelingfrom our thinking? For thatomission leaves us with the sterile, lifeless, mathematics most of us were taught at schooland that Western physical science adopted as its official language of authentication.Mathematics is, after all, just another human pursuit, a language, although a highlydeveloped and specialized one.

Perhaps one of the reasons why fascination with the primes and Riemann enigmas hastaken on such significance is because it turns out to be the key to some very profoundquestions - that is, questions of a kind we are discouraged from addressing or even

contemplating in our divided materialistic/mystical cultures. There are many welldocumented quotations from mathematicians speculatively alluding to this profundity.

In the text that follows its proposed that Riemanns Hypothesis can neither be proved nordisproved because of two key factors: first, a fundamental lack of understanding of whatthe prime numbers signify in the bigger cosmic context; and second, a discrepancy in thenumber system of the culture in which Riemann operated, and which still persists today,concerning the real number line and zero, again in relation to the bigger context.

Du Sautoy has said that mathematics is essentially ethereal and that the primes are"timeless numbers that exist in some world independent of our physical reality." That

would seem to be a promising starting place. Or is it still a no-go area, a taboo subject?

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1. The Threshold is...

In the following extract from a broaderThresholdwork, we focus on a few so called'irrational', ie mysteriously enigmatic, features of current orthodox mathematics. These

includepi, The Golden Ratio, The Fibonacci Series, Imaginary Numbers and The PrimeNumber Sequence. In so doing we find that there is a coherent pattern, an order withinwhich each of them performs an essential functional role.

For example, once it is realised, from the Thresholdperspective, that the apparentirrationality of theprime number sequence results from the merging of two simplesequences based on one consistent rationale, the obvious key question arises: Whatsignificant principles do these two sequences represent?And once we grasp what the prime numbers actually signify, it follows that Riemann'sintuitively sensed Hypothesis, unproven since 1859, is essentially sound butmathematically unprovable.

First: Overview + InsightOne way to resolve an apparently insoluble problem within any system is to bring to thewhole situation a sense ofpolarity. This involves gaining:(1) an overviewin order to see the system as a whole within its greater context, as well ashow it has evolved into its present state, and(2) maximum insightto penetrate to the effective centre, heart or core of the situation.Combining these two views enables us to detect the primary dynamics of the system andthus its essential meaning, what it's meantto be and do.

In this particular work, the fundamental human questions regarding who or what we are

in the individual and universal sense and what we're doing here are not addressed.They are, however, acknowledged as essentials of the bigger context in whichmathematics and all other disciplines find themselves.

Mathematics seems to be widely understood as the quest to discover ever clearer, moreaccurate and consistent patterns within this fragmented universe, to find order withinapparent randomness. As a code or language, it consists of various ways of ordering andquantifying information about the reality it serves to describe. It is, essentially, a mentaldiscipline which means it has the potential for endless abstraction.

Among the better known special features of this mathematical order would be such

values as those represented by the symbolspiandphi(The Golden Ratio), andsequences of numbers such as the Fibonacci series and theprime numbers.Among its most fundamental unresolved questions would be: 'What is actually meant bythe symbols forzero, one, and theso-called negative, irrationaland imaginary numbers?These questions inevitably lead to fundamentalpolarityissues such as: Which is primary:subjective or objective? Quality or quantity? Consciousness or matter?

The thresholdapproach, effectively, views current mathematics with its discretenumbers and consequent enigmas and conundrums from a higher dimension. Thisembraces the continuity, movement and polarity within the wholeness behind the frozenfragmentation, and enables us to see pattern and coherence not apparent from the lower

dimensional view. It's an extension of the principle by which some problems, insoluble intwo dimensions, can be solved in a three dimensional context.

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In contextIn the tradition of Einstein's mind experiments, there follows next a scenario, a model toserve as a context in which we can view mathematics as a system, a language, a code.How valid this is will show in how well it helps us resolve known problems and understand

ongoing observations. For present purposes the model is much simplified.It's also based on the following explicit assumptions.

Assumption 1: At theThresholdIn order to gain the power ofoverviewand insight, we need to mentally put ourselves atthe critical thresholdwhere the fundamental polaric forces of the cosmos, contraction andexpansionorgravityand levity, are in a state of dynamic balance. That is, at the border,the interface of the grossphysicalrealm and the subtle, pre-physical aethereal realm.

ThisAetherhas nothing to do with the 19th century, materialistic, so called luminiferous ether, theexistence of which was supposedly but falsely disproved by the Michelson-Morley experiments.More recently, the speculative notions ofdark energy, a zero point field, morphogenetic fields, thequantum vacuum and physicist Professor Paul Davies' quantum etherall seem to be pointing to theneed to re-acknowledge the ever-present, all-pervadingAether.

An introduction to this Aether is the 2006 bookAether - The Transcript ISBN 1-900034-10-7 and its partner, audio CD-ROMAether - Knowledge is Power. These includereasons why the Aether was written out of the script of Western science but has neveractually gone away. Meanwhile, in the view of UK professor of mathematics, MartinHuxley, most mathematicians are Platonists, and in Plato's time before the era ofmaterialistic science the Aether was very much a reality.

Ourthresholdis not a specific location in space, but has the lively, turbulent, transforming

characteristics of what some scientists call the zero point field. It's a kind of level, like thecritical temperature level at which water is transformed into vapour and vice versa. Itsessentialpolarityis acknowledged, albeit rather uneasily and in a negative way, byWestern science in, for example, its paired concepts of matter and anti-matter or gravityand anti-gravity.

Perhaps the most powerful benefit to be gained from awareness of this threshold is asimultaneous two-way perspective on the world. I can look outwards from my unique,physically located point of consciousness, my attention radiatingoutwards in anydirection. I also find that my consciousness can expand indefinitely so that I'm lookinginwards from all directions at once. Then I can choose to zoom in and focus on any

particular part of the world - a practical exercise in linking the absolute and the relative.

This inward seeing enables the expanded, 'greater I' to view the contracted, 'little I' in aswide a context as seems appropriate. We could liken this to viewing a whirlwind from thestillness found both high above it and at its centre, the eye of the storm - both qualitativelydifferent from the turbulence in between. We can also look at mathematics this way.

'Less is more' is an enigmatic yet common expression which carries the implicit notion ofpolarity. From the densephysicalperspective, less of the gross, physical-material meansmore of the subtle, mysterious 'something else' of the opposite quality, while from thesubtle aetherealview, less of that means more of the gross physical-material.

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Although we are considering apre-physicalera and level of reality, we can envisage ourscenario as an imaginary spherical bubble rhythmically contracting and expanding arounda focal centre-point of gravity. The dynamic of this fundamental polaric relationshipprovides a clue to the essential nature of what we might callpure orpotential energy.

Assumption 2: Polarity + our own presence =ThreefoldnessWe acknowledge our own conscious presence as an essential part of the whole situation,which therefore consists of (a) observer, (b) the act of observing and (c) that which isobserved. To ignore any of these three essential elements from our investigations wouldleave an incomplete, false picture, as is the case with the so-called 'objectivity' of Westernscience which persists with this unrealistic ideal, despite the contrary findings of Quantumphysics. We can, however, transcend the dilemma by explicitly acknowledging thepresence and unavoidable participation of the subject, ie the observer, who then has anobjective presence in the situation.

Scenario

Now to ourscenario. Lets first imagine a void consisting only of a universalconsciousness. Its first stage and level of existence before any manifestation is the pre-physical, continuously changing, indefinable, vital state of being, long known in theWestern world as theAetherwhich comprises various levels within its unity.Einstein explicitly asserted that space without the Aether is inconceivable, although he didnot require it for his theory of Relativity.

Within this aethereal medium or matrix is brought into being a specificpointof focussedconsciousness. The point automatically constitutes a centre around which theresimultaneously comes into existence a sphere. Resulting from the dynamic relationship,the tension, between its periphery and its centre point, this sphere constitutes a kind of

resonant cavity(RC) in which is generated a rhythmic pulsating ofcontractingandexpandingmovement, as waves of subtle, pre-physical energy.

In this polarity there seems to lie a clue to the resolution of the wave-or-particle dilemmaof modern physics. With point-centred gravitydrawing inwards and periphery-basedlevitydrawing outwards, the continuing tension represents a general tendency towards re-establishing an equilibrium state.

As the pulsating proceeds back and forth, standing waves (SWs) are propagated whichthemselves create ever smaller sub-spheres within the original. The result is increasingdensitytowards the centre point with raritytowards the periphery. The smaller the RC, the

faster the rhythm of resonance within it, ie the higher its pitch, note or tone. This wouldseem to be a potentially endless inward progression.

A distinction is drawn in this work between rhythm and frequency.Rhythm in a system implies continuous, subtle variation in the timing of movements dueto the ever changing circumstances, inner and outer.Frequency, although more convenient for comparing and calculating, implies a fixed,mechanically regular sequence of exactly equal timings. This means it's an idealised,abstract, average value. In this project we'll work with rhythm, since it's closer to reality.

The increasing density around the centre creates interference effects which inhibit theinward/outward wave movement. Eventually a critical level of obstruction or stasis isreached at which physical matterbegins to form first gaseous, then liquid, then solid.

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This all happens at the critical thresholdbetween the two distinct realms within the greateroneness. Matter, then, is amassed energy in an endless variety of forms, in a continuousprocess of disintegrating and being released/redeemed back into the ocean of cosmicpotential energy, theAether.

Taking a much longer view, we can see our original sphere as one of countless interactingspheres, all spinning and spiralling through a yet greater cosmos. The resultant distortingeffects of their combined influence give rise to the endless variety of physical forms andshapes of the physical-material world. Thus we have many partial forms orparts of thatone wholeness, ie quantityand diversitywithin the primal qualityofunity.

We can also see our individual human selves as points within that universalconsciousness, temporarily embodied in earthly matter formed around such centres, andheld in suspension at the critical border level, the threshold, between the two cosmic polarforces. The deeper such a point of consciousness has become embedded in solid, earthlymatter, as energy locked in gravity-bound inertia, the more it tends to interpret the world

around it accordingly - that is, as patterns formed by the separated bits and pieces in itssurroundings. This would seem to account for the present tendency to classify, count andcalculate, ie quantify, as a preferred way of gaining some control over our earthlycircumstances.

Given the specific focus of this particular work, ourscenario encompasses but does notdeal with the vast process of cosmic and natural evolution that is, from a rarefied'nothingness' , in strictly physical terms, to our present dense material universe.

The descent into matterThis scenario is a much simplified account of how consciousness, universal and

individual, makes what is sometimes called the 'descent into matter' - as fluidity becomessolidity, as movement becomes crystallised, as flow becomes frozen form.

Our model or scenario, then, is not like a mechanical structure of pre-assembled bits. It'smore like a stereogram image which, at first, looks like an amorphous mass of detail towhich we need to adjust our gaze. Then, if we 'get it', the main features of a recognisableform begin to emerge before our eyes... and suddenly it's there, visible as if in 3D where itwas not visible before, although nothing about it has physically changed.

So, taking the thresholdapproach, we don't start with numbers and symbols, which arethe end products of certain processes and mental activities. We start with the dynamic,fluid, moving energy processes of the cosmos which generate the world of material forms.Those numbers which orthodox mathematics labels irrationalare so called because theydo not fit into its own limited rationale. The labelling of other numbers as transcendentaldoes seem a more balanced approach.

2. EMU numbers / Threshold numbers

Equal Measure Unit (EMU) NumbersNormal systems of numbers are based on the assumption that in all circumstances within

a system, one specific unitis exactly equal in value to any other such unit, ie they arerigidly standardised. Well refer to these numbers as equal measure unit (EMU) numbers .

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Since humans have been able to reproduce standardised objects, such as bricks or coins,there seems to have been a strong temptation to look at the world around us as though itwere similarly made up of some kind of minute, identical building blocks. It's as thougheach of us, as an infant, was given an identical set of cubes which can be joined togetherin any number of combinations, and was taught that this is how everything in the world is

actually constituted.

We need to be aware, however, that this teaching does not coincide with the continuity,the fluid, ever changing, natural and cosmic reality of expansion/contraction, rarity/density,levity/gravity, or with how our consciousness or feelings function. The cosmos is not anassembly of cubic centimetres or any other such units. There is a fundamentalincompatibility here. The EMU number system has, nonetheless, proved a very usefulmental device when applied within the physical-material realm and within the range ofscales to which humans can readily relate.

However, the incompatibility becomes increasingly problematic nearer the extremes of the

human scale of comprehension, in both time and space. Here a growing sense ofmeaninglessness is experienced in trying to come to terms with extremely big andextremely small numbers. And pointing beyond the extremes of space and time we haveour vanishing limits, conveniently signposted as infinityand eternity.

All numbers are bi-polarBecause the aethereal realm is of a qualitatively different order from the physical-materialbasis of an EMU system, any alternative system would have to include such qualities asthe inclusive wholeness, continuity and fluidity of natural growth and transformation... fora start. Each number, including zero and the primes, consequently can be understood asbeing bi-polar in having two aspects:

(a) Within the number system, their strictly quantitative, exclusive, 'discrete bit'characteristics, and(b) Within the bigger cosmic picture, their qualities as elements in a continuous, all-inclusive reality in which 'prime number' means much more than just another numberwith some exceptional mathematical characteristics.

It follows that the expression 'generating prime numbers' has two distinctly differentmeanings.In (a) it means mathematically calculating them from other numbers.In (b) it means that there's a bigger ongoing generative/formative process in which thosevalues known as prime numbers are significant markers of proportion and resonance, as

we'll see later. So establishing the 'generating principle' is, essentially, not aboutmathematically eliminating certain numbers from a given sequence.

Meanwhile, we have at least become more aware of the limitations of any EMU system.

The grid filterOut of the rigid, precise EMU standardisation emerged what we might call a gridmentality. It serves as a kind of filter through which we look out at the physical world andassess it in terms of EMU cubes, squares, lengths, weights etc. And we are taught that allforms are assemblies of 'building block' particles, micro-digital, separate bits. In digitalcomputer graphics, 'bitmap'images composed of mosaic-like squares give a visual

example of this view, whether illustrating curves or straight lines.

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Such a conception is so convenient for measuring, counting, calculating and constructing- ie controlling - that it easily becomes a delusion, a false idea of how Nature and thecosmos actually are. This conception treats the end product of the cosmic formativeprocess as if it were the beginning. For example natural curves are seen first as made ofbits or points, rather than as the aftermath or trail of a curved movement pattern.

This kind of mindset seems to arise in those cultures which prioritise measuringandquantifying- as ifquantities are somehow more real than the qualities which we actuallyexperience first, ie as primary. The gridcan then become a trap, a prison, imposing rigidrestrictions on imagination, inspiration and intuition. It can also insulate us from the lessprecise, less easily controllable, fluid, fiery realm of our instincts, intuition and feelings.

In the attempt to dominate and control our world, the grid way of thinking has proved veryuseful. But it ignores the wider and longer term harmful consequences of such anexclusively quantifying attitude on ourqualityof life.

Zero and OneThe meaning ofzero-ness and one-ness go right to the heart of all questions aboutquantity and, therefore, mathematics. The way orthodox mathematics uses the symbolsforzero and one is symptomatic of the materialistic culture in which it has evolved.For example, the materialistic version ofzero signifies an empty space, an absence ofany material objects, while one signifies a single unit of something, as opposed to no suchunits or more than one.Such usage demonstrates binarythinking which, in its exclusive yes-or-no, on-or-offterms, fails to acknowledge that there is always a greater wholeness which includes allpairs of polar opposites. Binary thinking is the fundamental flaw at the root of all ourcurrent unstable digital technology.

Regarding zero, the thresholdview reveals a clearpolarity, inextricably linked with thenotions of infinityand the infinitesimal. We can conceive of an outwardinfinity of everexpanding/extending proportions disappearing into the beyond. Then there's its polaropposite, an inwardinfinity, a vanishing pointof unimaginable smallness, also known tosome as the infinitesimal. From our threshold we can sense these vanishing points orplanes asportals between the aethereal and physical realms, where the inward journeyreunites with the outward.

The infinitesimalseems to imply the idea of both the wormholes of the physicists and therabbit hole through whichAlice entered herWonderland(in a story written by a

mathematics lecturer). They also seem related to the quantum idea of the smallestquantifiable amount of energy and to the ancient Greek concept of the indivisible atom.And they offer a key to understanding the 'less is more' principle ofhomeopathywhichworks first on the aetheric level which then affects the physical body.

In the 17th century there was a historic dispute on this theme between Newton andLeibnitz which we can now see as apolarityissue. They were both reaching for thetranscendent level beyond the physical, but from polar opposite directions. Leibnitz,through the infinitesimal, emphasised consciousness which he expressed in thesubjective terms of our irreducible selformonad. Newton raised the level of thinking fromconcepts set in physical space to the more aethereal dimension oftime, with his momentswhichoccur in a context of continuous motion and flux.

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Zero, then, from the threshold, represents what we experience and recognise as atransitional moment or event. Viewed from the physical side, it's inherently paradoxical -the something somewhere between the EMU building blocks or between the boxes in agrid. We can, therefore, consider the symbol forzero either as representing a limitlessvoid full of potential in which no separate objects exist... OR as simply an empty gap-filler

in our decimal-based, EMU number system to denote that there are no units, tens orhundreds etc in their allotted places. Thus we have thepre-zero void and the nominalzero of the mathematical number system.

The numberone, can be treated as either representing a unity, a wholeness, a wholon,inclusive of any number of parts OR as a single, separate unitof something, an exclusivepart. This is the fundamental polarity ofwhole andpartwhich are always relative terms,ie in the manifest world there is always a bigger, inclusive whole/wholon and a smallerincluded part.

A wholeness, in this work, is always taken to be a relative phenomenon, a something with

its own identity which is not merely the sum of its parts. Yet it is temporary, transient andalso in continuous interflow with the greater wholeness of which it is but a part. Simpleexamples could be a piece of ice floating in water or a vortex formed in that water.

The Number LineFrom ourthresholdposition, the soundness of the traditional Number Line with a zero atits mid-point begins to look questionable. For here the space occupied by zero seems torepresent some kind of definite gap between -1 and +1, between the negative andpositive dimensions. By contrast, the year 1BC ran directly into year 1AD, with no 'yearzero' in between - only a critical, infinitesimalmoment of transition.

Since Western science does not yet openly acknowledge the universal presence of theall-pervading Aether, zero, implying a distinct empty space situated between thepotentialnegative numbers and the actualpositive numbers, is an artifice, an anomaly. This idea ofzero is somewhat similar to the idea (or ideal) in science of a so-calledperfect vacuum,something which doesn't physically exist in Nature and is therefore a meaningless term,except as an idea.

Such anomalies will inevitably arise in a frozen abstraction like the number line - that is,an abstraction from a dynamic, everchanging reality in which two polar opposite forcesare continuously and rhythmically interacting. Perhaps an updated version of theinfinitesimalwould be useful here.

Meanwhile, the central balancing point of a revised number line could be labelledminus/plus one. This point would serve as a portal for accessing the great void ofaethereal fullness implied by zero. And the inherent ambivalence of zero would then beacknowledged by the negative and positive numbers representing the aetherealandphysical realms. This balancing and cancelling out is pertinent to our reading ofRiemann's Hypothesis.

Negative Numbers and their derivativesThe word 'negate' means to deny the existence of something. Negative numbers express

the idea of something not manifest or present in a quantifiable form, as opposed to theactualmanifest presence of whatever it may be. That leaves a negative number as a

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potentialor virtual something. A debt, for example, is the idea of a potential payment of acertain quantity of something. A seed holds, in non-physical space, the potential physicalorganism it's destined to become, given the right conditions.This realm of an unmanifest, ie non-physical existence is the very nature of theAether.Mathematical formulations derived from a negative value can therefore be understood as

signifying the more subtle, potential, aethereal realm - the polar opposite of the positivenumbers which signify the more gross, actual physical realm.

The symbol i, Imaginary Numbersi,representing the square root of minus one, is a mathematical value much used inscience. It expresses a strange kind ofambivalence which we can now interpret in thecontext of the universal cosmicpolarity. For minus one, mathematically, is the product ofcombining - of multiplying together - minus one andplus one. This apparently anomalousoutcome would seem to be the result of the EMU number systems inadequacy for dealingin a wholistic way with the relationships of negative and positive numbers and with zeroand one.

Being the product of minus one and plus one places iright at the interface, the border, thethresholdbetween the numerically negative aethereal and thepositive physical. Andsignificantly, given the rhythmic pulsating of the cosmic polar forces, iis much used inengineering formulations concerning rhythm. From ourthresholdview, isymbolises thatcontinuously rhythmic and ambivalent state of the cosmos, with the two polar oppositeforces represented by the positive and negative numbers. Physics acknowledges this inthe idea of matter as vibrating energy patterns and anti-matter as its polar opposite.

Complex numbers are peculiar combinations ofrealand imaginarynumbers, part of thebizarreAlice through the Looking Glass world referred to by Professor du Sautoy, which

does seem like a kind of metaphor for our threshold.

Fractions (from the same root word as fragment) express the idea ofoneness dividedinto EMU parts, which then stand as separate entities within that oneness.

Cardinals / Ordinals and Ambivalence about countingThe two basic kinds of ordinary numbers are cardinals and ordinals.Cardinals represent manyness, how many there are of whatever is being referred to: 1,2, 3 etc. They imply a fixed, static, finite, condition, a state akin to the physical, spatial,material world of separate, exclusive entities.

Ordinalsrepresent akind of presence implying beforeness and afterness in a sequence:1st, 2nd, 3rd etc. They indicate a progressing, changing, non-finite process, more akin tothe unmanifest, living, aethereal realm, implying a continuous, inclusive flow of changingconditions or stages, both temporal and spatial.

The clear distinction of meaning between them has become con-fused in everyday usage,with cardinals being substituted for ordinals, which indicates the current dominance of thematerialisticmentality. For example:

9/11 refers to the 9th month and 11th day - whereas 24/7refers to 24 hours, 7 days.Chapter One is the 1stchapter, and Version 2the 2nd version and so on...

And the dispute as to whether the recent millennium should have been celebrated a yearlater than it was indicates an underlying lack of clarity about numbers and counting.

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Meanwhile, theprimes, as we'll see later, seem to serve as both ordinals and cardinals.

The Focus/radius LineBack with our original sphere, if we now envisage an imaginary line drawn radiating outfrom the centre point to any point on the peripheral plane, such a line would be whats

known as the radius of the sphere. Conversely, looking inwards from the peripherytowards the centre point, in the polar opposite direction, this line could also appropriatelybe called a focus, since it leads to the focal centre-point.

Pi: from sphere to cube, from circle to square across the thresholdWe can now see that there will always be a direct and constant relationship between thelength of the focus/radius line and other dimensions of the sphere it spans, as measuredin EMUs of length, squares and cubes. Translated into numbers, that constancy isexpressed through applying in various formulae the familiar, unresolvable numerical value3.142... represented by the symbolpi.

So, from ourthresholdview,piis the translator,converterorscaling factorbetween(a) the aethereal realm of the sphere and circle and (b) the physical-material realmof the cube and square.EMU calculations involvingpiwill always be approximations because of the fundamentalqualitative difference between the rarefied, spiralling curves of the aetherealrealm andthe dense, crystallised forms of thephysical-materialrealm. A hurricane can be frozenand reduced down in a picture to a pattern of quantifiable lengths, squares, cubes,velocities etc. But without including the contextof the converging atmospheric forces, wegain little understanding of the dynamics behind the phenomenon.

Phi: the Golden Ratio

The symbolphi, also known as the Golden Ratio orMean, represents a value which, it issuggested here, expresses a constant relationship of proportion between theunmanifest potential of the aetherealdimensionand ideal forms in thephysicaldimension. The golden mean is sometimes defined as the middle course between twoextremes.The numerical symmetry of 0.618 :1 and of 1:1.618 would seem to indicate a balancingpoint orthresholdin this 2-way cosmic polar relationship oftransformation. This may,however, vary at different levels of the rarity/density continuum in the cosmos, away fromor towards Earth's surface.Through the ages, humans have aesthetically appreciated this relationship ofproportionbetween the two realms as is widely evidenced in many sacred structures in

architecture, art and design, for example in ancient Greece. This would seem to bebecause forms which are proportioned according to these ratios resonate harmoniouslywith the proportions inherent in our our own structure.

The Fibonacci SeriesThe Fibonacci Series is a variation of the relationship represented byphi, it issuggested here. However, it only applies to the material level of existence, after matterhas come into being and become the world of natural organic forms. As such it is mainlyapplicable in the outwardlyextending direction, from central-physical density towardsperipheral-aethereal rarity; ie in growth. And, as a numerical series, it only begins toconverge closely with the Golden Ratio after several stages in its progression.

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Both thephiand Fibonacciseries can be expressed geometrically as a particular kind ofspiral, each of which can be seen to generate a series of rectangles. So here we have ademonstrable connection between the aethereal and the physical in the continuity of aspiral, a critical line, spawning a series of separate quantifiable forms.

The Zeta FunctionThe zeta function is a mathematical device first applied to the primes by the greatmathematician, Euler. It was later elaborated by Riemann. The simpler version producesan infinite series of decreasing reciprocals from 1, 1/2, 1/3, 1/4... when x, signifying the'power' of the denominator, equals 1. It coincides with the musical Harmonic Series, firstestablished by Pythagorus,and as such is relevant to ourinwardprogression ofdiminishing RCs. It also shows the primes to be an infinite series.

From the threshold perspective, the zeta function symbolises the pattern of a real andfundamental cosmic process. It concerns the series of proportional resonances set up inthe formation of the physical universe. However, when mathematicians, in an abstract

way, speculatively substitute different values for those proportions, the results may or maynot bear any meaningful relation to the reality being symbolised.

One significant outcome of Euler's efforts here was the discovery of an unexpectedconvergence ofaddition and multiplication in the primes. These were found to exhibit bothfunctions simultaneously, implying that they have a dualnature. Seen from ourthreshold,multiplication, as in geometricprogressions, is about the quality ofproportion which isessentially aethereal, while addition, as in arithmeticprogressions, is strictly about EMUquantities which derive from the physical-material realm. This would suggest that theprime numbers represent those values where the aethereal and physical realms coincide,where they balance or cancel out into a kind of equivalence or zero value.

Threshold numbers, then,are those values includingpi,phi andi - which enabletransition and transformation across the threshold we have been considering.

3. The Primes & Riemann

One basic questionGiven their unique numerical properties, what does the peculiar and apparently irrational

sequence of the primes signify?

Three questions mathematicians have long been asking about the primes1. How many primes are there up to any point in the normal Number Line?2. Can Riemanns Hypothesis be conclusively, mathematically proved?3. Is there a mathematically sound formula for predicting the next prime number?

Riemann: What's the relevance?Riemanns abstract, artificially constructed 'landscape' (see du Sautoy's book) is no morethan an elaborate graph, a visual illusion produced by combining three numerical grids. Itdoes, nonetheless, reveal a certain consistency of pattern in the numbers. And this has

long been felt to be basically sound by many mathematicians - so much so that theories

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have been constructed on the assumption that it is so, despite the continuing absence ofany conclusive proof after a century and a half.

The way that it seems to convey a deep, intuitive truth somewhat resembles how dreams,archetypal forms and artistic works have their effect, ie at a non-intellectual level of our

consciousness. Meanwhile, the quest for a 'perfect' proof illustrates a fundamentaldifference between the more idealistic approach of mathematicians and the morepragmatic approach of scientists, ideas and ideals being essentially aethereal.

Cultural assumptionsIts suggested here that the problem has remained unresolved because of certainunsound but unquestioned assumptions in the culture behind Riemann's thinking,concerning both mathematics and bigger cosmological issues. For there does still seemto be an irresistible tendency to reduce the significance of the greater reality indicated bythe primes and by Riemann down to EMU mathematical terms and reasoning. And thiswould seem to be why the primes as a sequence have not been found to reveal a clear

EMU pattern. They seem to tell of something else entirely, something beyond numbers.

Two-faced primesOurthresholdview reveals that the primes have their own inherentpolarity: they have twoaspects, so to speak, as Euler's Productshowed. A crude analogy would be a valve orduct connecting two distinct spaces and therefore simultaneously part of both.

1. In the aetheric, pre-physical, order and formative process, the primes serve asordinals, simply 1st, 2nd etc with no EMU number values, while...2. Within the EMU 'real' number line, they stand as cardinals with distinct number values.These two aspects of the primes are qualitatively different. So their connection cannot

be established by direct calculation, only indirectly by inference.

The cosmic fretboardLet's return to our imaginary focus/radius line between the periphery and the centre of thecosmic sphere, our original RC. Moving inwards, the ever closer together standing waves(SWs) intersect our line at ever shorter intervals. The intersections mark out the peripheryof each smaller, concentric sub-sphere contained within it, each resonating at acorrespondingly faster rhythm or higher pitch.

Now let's imagine our line to be a string stretched along the neck of an imaginary musicalinstrument whose hollow body is the original RC. A series of frets along the neck mark out

the proportionate distances along the vibrating string for producing particular notes withina harmonic order, which is done by by pressing and thereby shortening the string. Aharmonic order means a series of notes in harmonious resonance with the onefundamental tone of the original RC.(Physical soundis quantifiablewhereas the quality oftone is aetheric.)

Significantly, only at certain frets can we produce what are known as 'harmonics'. Theyare the more subtle sounding notes produced by lightly and momentarily touching thestring above those frets while it is being plucked, so that the whole string vibrates inharmony with its simultaneously shortened length. That is, instead of just the shortenedlength vibrating as a result of the string being pressed down. These special fret positions

would seem to parallel where our focus/radius line is intersected by the RCs.

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At all other possible positions - or sizes of RC - the result would be a cancelling out ofresonances, which leaves us only the prime/harmonic positions. This direct relationshipbetween the resonance of particular shorter lengths and the whole length of the stringparallels the way that prime numbers are only divisible by, ie are resonant with, oneness.

Regarding harmonic systems, the Western 12 note, even tempered scale is an artificialcreation of an EMU, 'building block' culture. It has proved very useful and creativelyempowering in many ways. However, it has also inhibited the potential for some subtleand deeply expressive music only possible when not restricted by such a 'grid' system.

Inwards/outwardsThe fundamental rhythmic alternating between periphery and centre produces twoseparate and differing sequences of SWs and their resulting RCs. These are the inwardand the outwardsequences. Because of the increasing density towards the centre, theimpulses or waves of the outwardsequence are more obstructed at their outset than thewaves of the inwardsequence.

This would seem to account for the discrepancies between (a) the ratios of the Fibonacciseries at its lower number values, close to the centre, and (b) the constant value of theGolden Ratio (phi) - until they begin to coincide more closely after about a dozen steps ofthe Fibonacci series.

So the sequence of prime numbers consists of two progressions going in polar oppositedirections but merged into one series of EMU numbers - ie two mutually opposed sets ofstatistics. This is not the same as, but somewhat like, superimposing two sequences oftemperature changes, both set out in order of lowest to highest values. However, onerepresents a phase offallingtemperature and the other a phase ofrisingtemperature,and when combined, the two number sequences do not exactly coincide. The result is an

apparently irrational series.

The primes are...Bringing together our two lines of approach, we recognise how the coincidence of themathematical zeta function (when x = 1) with the musical harmonic series, offers a crucialclue linking the origins of the cosmos, the so called music of the spheres andmathematical logic. We recognise how, in EMU terms, the progression of numbers knownas the primes corresponds only to those cosmic 'frets' which produce the harmonic seriesof notes resonant with the one original RC, the whole cosmos. As numbers, they are onlydivisible by the numberone.

The primes thus symbolise irreducible wholes, wholons, complete integral forms. Theyare like bubbles or eggs whose proportions express the balancing of inward and outwardforces, and that zero level is represented by the surfaces or planes that are theirsurrounding membranes. Resonance, at particular rhythms, can create and disintegrateforms, and physical science deals extensively with the rhythms or frequencies of waves.The primes, then, represent a precisely proportioned series of critical rhythms.

So, a pattern of progression begins to emerge.- It starts at the aetherealperiphery with the first and biggest number of all which is one.- It progresses infinitely inward, with the ever decreasing sized sub-spheres in ever

closer proximity, as their number rises. At the centre, the smallest number is one.

- The process involves two-way rhythmic movement.

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However, as we move into the extremely large numbers and the fast vibrational rhythmsof very small RCs, it all becomes inconceivable, inaudible, meaningless and irrelevant toour human scale of experience and appreciation. Which raises the question as to whatbenefit there might be in trying to extend the sequence of primes ad infinitum - exceptperhaps as an exercise in obsessive counting or for devising hard-to-crack security

codes. And both of these seem like clear signs of an endless spiralling off into absurdity.

4. Beyond mathematics Riemann's significance

Some things are beyond mathematicsIn the 1930s mathematician, Kurt Godel, established that some axiomatic principlescannot be proved or disproved from within the existing discipline of mathematics.This seems to apply to the many attempts to prove or disprove Riemann's Hypothesis.The 'problem' of the primes, therefore, can only be comprehensively tackled from outsidethe confines of current mathematics. Otherwise we find ourselves locked into a vainsearch for some mathematical 'holy grail'.

In the bigger context which ourThresholdview brings, we can now appreciate an overallpattern of principles governing several previously enigmatic, and so called 'irrational',features of mathematics. We find meaningand consistency, as opposed to merelyuniqueness and utilitarian value, inpi,phi, the Fibonacciseries, i and the primes. Othersignificant numerical values can also perhaps now be re-assessed on this basis.

The solutions, as we have found withpietc, do not lie in the numbers themselves.Consequently, we've extended our thinking beyond the current limits of orthodox

mathematics, and through this approach have become aware of some of the limitations ofthe EMU number system.

By extending the conceptual context within which it is viewed, we have foundRiemann's Hypothesis to be essentially correct. That context is now summarised.

(A) Polarity-within-wholeness (the governing principle)Consciousness/matter, rarity/density, aethereal/physical, levity/gravity, outward/inward,expansion/contraction, radius/focus, negative/positive, Golden Ratio/Fibonacci series,ordinals/cardinals, resonant cavities/prime numbers ...

(B) The formative processPulsation between periphery and centre of an original cosmic sphere, RC1, creates

standing waves which form, within that original one sphere, a series of concentricsub-spheres, RCs which are in resonance with it alone.

The RCs - in order of decreasing magnitude and faster rhythmic resonance -correspond to the musical Harmonic Series (HS).

The HS is known to coincide with the mathematical Zeta Function (when x=1).The Zeta Function (when x=1) is an infinite series starting from one.

The HS is, therefore, an infinite sequence of resonances, its fundamental (lowest)tone being that of the original one RC.

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Primes are numbers mathematically resonant only with number one, and all EMUnumbers can be reduced down to combinations of primes.

Thus theprime numbers in their dual role, are:

(a) a series ofordinals representing the infinite inward sequence of RCs which areresonant only with the original RC1, and(b) those cardinalnumbers divisible only by themselves and the number one.

Working definition (of the primes)The primes, expressed as EMU numbers, represent a scale of measures of the cosmicformative process, between the potential unformed aethereal realm and actual physicalform, through resonance. As such, they constitute a bi-polar cosmic number system.Riemann, it is suggested here, intuitively sensed the universal threshold between the twopolaric forces of the cosmos and expressed this balance, through the language of

mathematics, as his critical line of zero values.We are now adding that:1. Only in theprimes is harmonic resonance 'crystallised' into EMU number values.2. Phiand the Fibonacci Series are scaling measures of that transformative process.3.The zeta function (when x = 1) expresses the unfolding sequence of resonances.4. Imaginary numbers express the balancing and cancelling out between the

negative/potential/aethereal and the positive/actual/physical realms.

So Riemann's inspired assertion that the primes 'very probably' represent amathematically consistent progression, demonstrable as an infinite line of zeros, hasproved sound. The essence of his hypothesis has been shown to be correct, and

our reaching this conclusion has involved no mathematical calculations. Any subsequenttechnical calculating and formulating is left to the experts in this field.

On the level ofintuitive inspiration being expressed in limited physical-material terms, wecan see some parallels with, say, Einstein's curved space, Newton, the alchemist, andeven Charles Darwin. For Darwin did put his apparently materialistic evolutionary theoryin a much more open ended context when he said I am inclined to view the world as if itwere the result of designed laws but with the details left to chance.

Proof = ?The question of what constitutes a validprooffor whom is whole other subject in itself. But

by the standards of the Clay Mathematical Institute (USA), ourthresholdapproach doessignificantly 'shed light on... the mysteries surrounding the prime numbers', and has foundthat their'distribution among all natural numbers does follow a... pattern'such that thenext one may be predicted.

The primes, six-ness and the 'twins'All the primes can be seen from ourthresholdto occur within a consistent numericalpattern - in which the 'perfect' number6 is significant. The number 6 is called 'perfect' inmathematics because it's factors, 1,2 and 3, when either added or multiplied together,give the same result, 6. This combining of the functions of addition and multiplication isalso, significantly, characteristic of the role the primes play in the zeta function.

And there's a clue in the old trick: Take any prime greater than 5, square it, add 17, divideby 12, and the answer will give a remainder of 6, or on a calculator .5 (1/2 of 12).

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The significance of six-ness, beyond the merely symbolic mathematical statement(1+2+3) = (1x2x3), will be dealt with in a later publication.

The pattern in the occurrence of the primes, within the EMU number system, is as follows.There are two sequences of numbers, representing the two polar directions described

earlier. One of these starts with the void which for now we're callingpre-zero, and theother starts with 1 or 7. Each is an arithmetic progression with 6 as the increment, whichgives two sequences: pre-0, 5, 11, 17, 23, 29... and 1, 7, 13, 19, 25, 31...The three factors of 6 are excluded, if it's assumed that 1 is not a prime number.

When the two sequences are set out in parallel, the so called twinningcan then be seenas an integral part of the pattern. And as the multiples of each new prime are alsoeliminated and the two sequences are merged in numerical order into one, we are leftwith the familiar, infinite prime number sequence. So 'predicting' the next prime in thesequence eventually becomes a matter of deciding at which point it's no longer viable, recomputer power, cost etc, to keep eliminating the ever increasing number of non-primes.

From the meta-mathematical, thresholdperspective of a cosmos in continuous change,six-ness can be understood as a unique quality of the cosmic transformative process. Itrepresents each 'cycle' of a vortex traced out in the progressive formation of RC spheres.Why has a proof of Riemann's Hypothesis proved so intractable?First, we have to acknowledge that mathematics freezes and reduces the fluid, mobilecontinuity of our cosmos down to a grid-like, 'building block', EMU system. This is theartificial abstraction within which Riemann was working. And within it no qualitativedistinction seems to have been made between the ordinalseries (of RCs) with which theprimes correspond and the occurrence of the primes in the EMU number line ascardinals.

Second, the idea of a mathematical proofonly has meaning within that system, given allits implicit assumptions of permanence and sameness etc. For within the bigger contextof our ever changing cosmic environment, nothing remains truly the same as it was.Thus the inconsistencies of mathematics, a few of which are here briefly addressed.

1. We find we are trying to establish hard, precise, ie finite information about what hasbeen officially designated an infinite series. And we are attempting to do this from withinthe finite limitations of an EMU system that assumes infinityto mean an endlesslyoutward, expansive progression. The primes, however, represent an infinitely inwardprogression. Bearing in mind the logarithmic spirals of the Golden Ratio and Fibonacci,this also touches upon the classic conundrum of trying to reconcile in exact numbers thecube and the sphere, the square and the circle, the straight line and the curve.

2. The Number Line, viewed from ourthreshold, is an abstract, artificial, linear-thinkingconstruction. What it presents as the negative scale of numbers is a mirage, a projectingof the physical-material world ofpositive numbers on to the unmanifest, potential realm, iethe Aether. And this projection seems to assume falsely that the pre-physical aetherealrealm can validly be portrayed in materialistic EMU terms.

3. The position on the Number Line, labelled +1/2, at which Riemann's transverse line of

Imaginary numbers intersects, turns out to be a misleading misconception. It sits midwaybetween zero and one. But the zero on the line, as mentioned earlier, is essentially an

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indicator of the interface between the negatives and the positives, between -1 and +1. Sosome adjustment to Riemann's Number Lines is required.

Imagine a new number lineIf the north/south line is re-positioned to intersect the Real Number line at the point

midway between -1 and +1, at what was the zero point or what we could now call'minus/plus 1', our picture starts to gain some coherence. For the interaction (ormultiplying together) of -1 and +1 at that point produces -1. And i, the 'square root' of 1,is the first of the Imaginary Numbers which are now at their new place, minus/plus 1, onthe north/south line intersecting our revised Real Number line. This transverse line iswhere all the negatives and positives meet and mutually cancel out each other.

As such, it represents the plane where the aethereal(negative) andphysical(positive)realms interface and are in balance. That plane can be represented in 2D as an infiniteline of zeros, reminiscent of what Riemann seems to have intuitively envisaged, butwhich he expressed in a more complicated way involving complex numbers.

It would also seem to represent the thresholdwhich this work has been highlighting andits bigger context, ourscenario of a polaric, rhythmically pulsating, resonating cosmos.Riemann had the inspired idea for an ingenious mathematical structure which, inprinciple, was sound but had an inherent design flaw. And since 1859, mathematicianshave been analysing it and speculating about it without, it seems, realising there was afundamental problem of misalignment in the underlying culture and therefore in hisdesign. Consequently, his hypothesis has remained neither proved nor disproved.

A practical approachInstead of starting with the abstract numbers and formulae Riemann finished up with, let's

assume that there do exist, at some level of reality, phenomena which are expressible inthe mathematical language of his hypothesis. A sequence or row of zero values suggestsperhaps a line or plane resulting from the balancing and cancelling out where two forcesmeet. That line could be straight or curved, or perhaps a spiral.

In ourthresholdview of the inward/outward pulsation between the aethereal-peripheraland the physical-centric realms, two similar but not identical spirals which track thatprocess come to mind: the Golden Ratio and the Fibonaccispirals. These each generatea series of rectangular figures which represent the 'crystallising' of moving, aetheric,potential energy into physical-material form. And the rectangles are generated at a rateproportional to the turning of the spiral. Thus is produced a progression of quantifiable

forms from continuous movement.

The rectangular forms could represent our resonant cavity (RC) spheres, now contractedand collapsed under the force of an excess of gravity over levity. These polar oppositeaspects of the one process would be expressed in two ways:1. A continuous spiralling of varying proportions, representing the aethereal dimension.2. A series of discrete steps or numbers, representing the physical dimension.

A correspondence has already been established in mathematics between stages of acontinuous progression of harmonic resonance and a series of decreasing reciprocalnumber values - in Euler's Zeta Function. This was later made more complicated by

Riemann when he introduced complex numbers which combine realand imaginarynumbers. Here we reach the heart of the puzzle concerning his famous hypothesis.

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Riemann's misaligned bridgeLet's think this through in a practical way, but not get too stuck on concrete images. Wecan imagine Riemann conceiving of a bridge to make a connection across some kind ofdividing line of which he was intuitively aware. But the two halves of the bridge ended up

misaligned where they were intended to meet. That was because, unbeknown to him, thetwo worlds on either side each had a different understanding ofzero. And thatincompatibility came out in his calculations and design as a shift of + to the east on therealnumberline. (Remember, this is only an analogy.)

So Riemann then had to introduce an add-on corrective to cancel out the unfortunatemisalignment and complete the bridge. Mathematically, the ambivalent nature ofImaginary numbers, which combine the positive and negative and so mutually affect bothsides, served this purpose and helped restore the missing alignment - that is, his line ofzeros. A simpler 'bridge' could have been built long before, using Euler's Zeta function,had the pattern and meaning of the primes sequence and the fuller meaning of zero been

recognised at the time.

The long term unquestioning acceptance of Riemann's understandable but flawedassumption brings to mind the words of Christof, the arch media manipulator in thatinsightful film The Truman Show, when he says: 'We accept the reality of the world withwhich we're presented. It's as simple as that.'

The prime numbers sequence has also been found to have correspondences with the socalled quantum realm. Which is not surprising when it's realised, from the thresholdperspective, that the quantum world, with all its unfamiliar, non-physical characteristics, isan aspect of the pre-physical Aether.

So, what vital factor is missing that makes mathematics such a turn-off for somany?The brief answer is: What's missing is one side of the great cosmic equation, that is, theequation which expresses the balancing out of the inherent polarity throughout our wholecosmos, and all within it, towards a state of equilibrium.

All power, energy and forms arise out of the rhythmic interplay of the universal polaricforces of the cosmos. And all force must originally be exertedat some level of existenceby some motivating source or act of will (which is not to imply any religious connotations).

In recent centuries the objective, physical-material aspect of our world has been studiedin great detail by scientists, but to the neglect of consciousness and the aetherealdimension. So, regarding mathematics, the missing half is all about how we subjectivelyexperience and appreciate the qualities of quantities, as opposed to how we merelyname, count and calculate quantities in an objective way.

Archimedes and PlatoRecently, a UK professor of mathematics responded to an earlier version of ThresholdMathematics by quoting Archimede's famous statement about leverage, 'Give mesomewhere to stand, and I will move the earth.' His point was that this equally applies to

changing our perspective such that everything 'shifts' and we gain a whole new

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understanding.The professor also said he thought most mathematicians are platonists,and in the collective consciousness of Plato's time theAetherwas very much a reality.

Disconnected = polarisedFrom ourthresholdview, a general lack in a culture of any deep, direct conscious

connection with its vital cosmic environment suggests that there'll surely be some kind ofresulting split into binary extremes. In the rarefied mathematical world of dedicated primesspecialists, there does seem to be both an extreme focussing on highly complexcalculating and an unlimited expansion into ever more elaborate and obscure abstraction.

Assumptions not questionedThe wholeprimes/Riemann saga seems to have arisen from a lack of willingness or abilityto question some basic assumptions behind what is taught as mathematics. The result ofthis compliance is one huge cloud of abstract thoughtforms hovering over mountains ofspeculative papers, the collective attempt of many agile and creative minds to rediscoverthe longed for coherence, elegance and clarity, intuitively knowing it's there but

inadequately equipped for the job.

The advent of the binary, digital computer hugely boosted mathematical pursuits, both inscale and speed. But perhaps, in the rush, other ways of thinking, not reducible to binarylogic, have become further neglected resulting in an even more distorted imbalance.Constructing a system, a world, of mathematics with EMU building blocks and grids hasproved very useful and also an endlessly fascinating subject in itself for some. However,to expand mathematics beyond the limits of that constricting worldview would require amuch more flexible, organic and aethereal system of concepts and thinking.

5. The Aether and mathematics

Whatever happened to the Aether?Has the indoctrination process known as education rendered us unable or unwilling toseriously consider the possibility that the timeless, universal Aether does in fact exist?If either, then we are left with the incoherent multitude of substitutes already indicated.Even the highly acclaimed systems theorist and writer, Ervin Laszlo, affirms the Aether inhis 2004 book Science and the Akashic Field, inwhich he treats akasha as a straightsynonym for the (A)ether.So it's neither something religious nor materialistic. But in this cleverly divided and ruledworld, it's what both of those entrenched camps lack. The challenge seems to be moreone of unlearning redundant habits by freeing up our thinking and intuition than of addingon more complication.

What is it about mathematics?Only when we stop and think about it do we become aware of just how steeped we are inthe materialistic culture of quantifying and numbers. We've been absorbing it from ourinfancy when we would have had no awareness of this process going on.Certainly, any childhood talent for numbers and calculating is rightly to be encouraged.But without a counterbalancing development of our sense of wholeness, continuity, qualityetc, our thinking becomes distorted - and all the more so when this over-emphasis onanalysis at the expense ofsynthesis is rewarded with praise, prizes and certificates,

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leading to career and financial advantages. We're conditioned to admire cleverness andto ignore or devalue wisdom.

The cold, sterile, precise, abstract qualities mostly associated with mathematics evidentlyhave their appeal to a small minority - as a pure, detached, feeling-free discipline. Yet the

obvious passion with which some dedicated mathematicians seek to discover new truthsand resolve mysteries does seem to speak of something much more than mathematics.Marcus du Sautoy refers to a high proportion of musicians in the world of mathematics.

However, becoming fascinated, captivated and bewitched by the endless possiblepermutations it offers can lead to a 'not seeing the forest for the trees' - as well as to ourconfused culture of endless statistics and contradictory counter-statistics. It's as if there isa heartfelt, intuitive knowing and longing for a deeper understanding of this world. Yetmost seem trapped in the grid-view, building block mindset, within a career structure andculture that eventually sabotage that driving, creative urge. The frustrated fascination withthe prime numbers and Riemann's Hypothesis, as a number of popular books and various

websites confirm, seems to be a case in point.

It's here that we clearly see in action the dubious assumption that 'The answer lies in thenumbers' - anapproach taken to painfully obsessive extremes in Darren Aronofsky's filmPi. For the central character, a mathematician, numbers took on a hugely inflatedsignificance, amounting to a false mantle of reality.His quest was as futile as trying to calculate and predict the ripples in a pond by 'freeze-framing' a picture of the pond first - which means you've already lost its true nature andare dealing only with an abstraction.

6. PostscriptIn animated suspensionSo ourThresholdview and scenario are perhaps not merely fanciful ideas, but do providea useful perspective on where and how we human beings are in the bigger picture. Wecan now envision ourselves held in animated suspension between the subtle, expansive,rarified aetherealand the gross, contractive, densified,physicaldimensions - somewhatlike bubbles floating between the elements of air and water.

Without the wholistic view from the threshold, the data - numbers, in the case of

mathematics - sometimes seem to have no coherent meaning despite obviousconsistencies and patterns. And although some people may prefer to use other wordsinstead of Aether - for example, dark energy, the quantum vacuum, quintessence orvarious fields, we'd still be essentially reaching towards the same thing.

Intuition + intellectAt the thresholdwe transcend the subjective/objective divide through identifying ourindividual consciousness with the universal consciousness. This is an intuitive mentalprocess beyond intellectual, rational or computational operations. And since we are hereventuring into realities beyond the physical-material realm, certain precautions arerequired. Any conclusions thereby reached must be subjected to intellectual scrutiny to

test their practical validity. In the present work, the rational consistency of the numbersarrived at provides such validation.

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Mathematical models- how real?We also need to beware of abstract mathematical models, when employed as tools for

scientific investigation into the real world. For here there can be a strong temptation tomake observations fit the model which, as an attractively self-consistent approximation,may become an item of unquestionable dogma, an established but false fact or theory.Presently, two contenders for this category might well be dark matterand dark energy.

Test of consistencyFrom ourthresholdwe can also envisage ourselves being tested to see how deeply ourconsciousness has been influenced by being embedded in dense, static matter suchthat some now believe that matter is the ultimate reality rather than consciousness,despite accepting since Einstein that matter is condensed energy. And energy itselfremains an enigma until the cosmic levity/gravitydynamic is acknowledged.

New thinking, new termsInevitably, the thresholdperspective requires some new mathematical terms and symbolsto handle its extended vision of wholeness and polarity, but that's beyond the scope ofthis limited paper. And finally, because this thresholdview cannot be reduced down andforced to fit into the narrow confines of the mechanistic-materialistic, 'boxed in' way ofthinking, that does not invalidate it in the greater context of humanity's evolvingconsciousness.

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